A bailiff called for my ex, can he demand to see my council tax bill?

Yes

When a bailiff is shown evidence the debtor does not live at the address, he is unable to take control of goods there,^[1] and must withdraw from the property with all due reasonable speed. Otherwise the bailiff is liable to pay the occupant the sum of £10,000.^[2]

^{[1] Paragraph 14(6) of Schedule 12 of the Tribunals, Courts and Enforcement Act 2007
[2] Neil Tunstall v Direct Collection Bailiffs Ltd and Hussein}